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Theorem xpiderm 6106
Description: A square Cartesian product is an equivalence relation (in general it's not a poset). (Contributed by Jim Kingdon, 22-Aug-2019.)
Assertion
Ref Expression
xpiderm (x x A → (A × A) Er A)
Distinct variable group:   x,A

Proof of Theorem xpiderm
StepHypRef Expression
1 relxp 4389 . . 3 Rel (A × A)
21a1i 9 . 2 (x x A → Rel (A × A))
3 dmxpm 4497 . 2 (x x A → dom (A × A) = A)
4 cnvxp 4684 . . . 4 (A × A) = (A × A)
5 xpidtr 4657 . . . 4 ((A × A) ∘ (A × A)) ⊆ (A × A)
6 uneq1 3084 . . . . 5 ((A × A) = (A × A) → ((A × A) ∪ (A × A)) = ((A × A) ∪ (A × A)))
7 unss2 3108 . . . . 5 (((A × A) ∘ (A × A)) ⊆ (A × A) → ((A × A) ∪ ((A × A) ∘ (A × A))) ⊆ ((A × A) ∪ (A × A)))
8 unidm 3080 . . . . . 6 ((A × A) ∪ (A × A)) = (A × A)
9 eqtr 2054 . . . . . . 7 ((((A × A) ∪ (A × A)) = ((A × A) ∪ (A × A)) ((A × A) ∪ (A × A)) = (A × A)) → ((A × A) ∪ (A × A)) = (A × A))
10 sseq2 2961 . . . . . . . 8 (((A × A) ∪ (A × A)) = (A × A) → (((A × A) ∪ ((A × A) ∘ (A × A))) ⊆ ((A × A) ∪ (A × A)) ↔ ((A × A) ∪ ((A × A) ∘ (A × A))) ⊆ (A × A)))
1110biimpd 132 . . . . . . 7 (((A × A) ∪ (A × A)) = (A × A) → (((A × A) ∪ ((A × A) ∘ (A × A))) ⊆ ((A × A) ∪ (A × A)) → ((A × A) ∪ ((A × A) ∘ (A × A))) ⊆ (A × A)))
129, 11syl 14 . . . . . 6 ((((A × A) ∪ (A × A)) = ((A × A) ∪ (A × A)) ((A × A) ∪ (A × A)) = (A × A)) → (((A × A) ∪ ((A × A) ∘ (A × A))) ⊆ ((A × A) ∪ (A × A)) → ((A × A) ∪ ((A × A) ∘ (A × A))) ⊆ (A × A)))
138, 12mpan2 401 . . . . 5 (((A × A) ∪ (A × A)) = ((A × A) ∪ (A × A)) → (((A × A) ∪ ((A × A) ∘ (A × A))) ⊆ ((A × A) ∪ (A × A)) → ((A × A) ∪ ((A × A) ∘ (A × A))) ⊆ (A × A)))
146, 7, 13syl2im 34 . . . 4 ((A × A) = (A × A) → (((A × A) ∘ (A × A)) ⊆ (A × A) → ((A × A) ∪ ((A × A) ∘ (A × A))) ⊆ (A × A)))
154, 5, 14mp2 16 . . 3 ((A × A) ∪ ((A × A) ∘ (A × A))) ⊆ (A × A)
1615a1i 9 . 2 (x x A → ((A × A) ∪ ((A × A) ∘ (A × A))) ⊆ (A × A))
17 df-er 6035 . 2 ((A × A) Er A ↔ (Rel (A × A) dom (A × A) = A ((A × A) ∪ ((A × A) ∘ (A × A))) ⊆ (A × A)))
182, 3, 16, 17syl3anbrc 1087 1 (x x A → (A × A) Er A)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242  wex 1378   wcel 1390  cun 2909  wss 2911   × cxp 4285  ccnv 4286  dom cdm 4287  ccom 4291  Rel wrel 4292   Er wer 6032
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3865  ax-pow 3917  ax-pr 3934
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3352  df-sn 3372  df-pr 3373  df-op 3375  df-br 3755  df-opab 3809  df-xp 4293  df-rel 4294  df-cnv 4295  df-co 4296  df-dm 4297  df-er 6035
This theorem is referenced by: (None)
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